Unique metric segments in the hyperspace over a strictly convex Minkowski space

Let (ℝn, {double pipe} ̇ {double pipe}B) be a Minkowski space (finite dimensional Banach space) with the unit ball B, and let ςHB be the Hausdorff metric induced by {double pipe} ̇ {double pipe}B in the hyperspace Kn of convex bodies (compact, convex subsets of ℝn with nonempty interior). Schneider (Bull. Soc. Roy. Sci. Li'ege 50:5-7, 1981) characterized pairs of elements of Kn which can be joined by unique metric segments with respect to ςH-the Hausdorff metric induced by the Euclidean norm {double pipe} ̇ {double pipe}Bn. In Bogdewicz and Grzybowski (Banach Center Publ., Warsaw, 75-88, 2009) we proved a counterpart of Schneider's theorem for the hyperspace (K2,ςHB) over any two-dimensional Minkowski space. In this paper we characterize pairs of convex bodies in Kn which can be joined by unique metric segments with respect to ςHB for a strictly convex unit ball B and an arbitrary dimension n (Theorem 3. 1). © 2012 The Author(s).